Showing a group is isomorphic to a group with known presentation

Question

Let $H$ be a group with presentation $\langle h_1, \dots, h_n \mid r_1 = \dots = r_m = 1\rangle$. If there are $g_1, \dots, g_n \in G$ which satisfy the relations $r_1, \dots, r_m$, when is $\varphi : H \to G$ generated by $\varphi(h_i) = g_i$ an isomorphism? In particular, is there an effective way to check that the elements $g_i$ don't satisfy any additional relations?

Answer 1

If we could effectively to check that $g_i$ don't satisfy additional relations, then we can decide whether $G\cong H$. But the problem of isomorphism is undecidable.